Question

prove that BD×CD = AD^2​

Answer 1

Answer:

In triangles ABC and DBA:

• ∠ABC = ∠DBA  (common angle)
• ∠BAC = ∠BDA  (both are right angles)

Therefore triangles ABC and DBA are similar.

In triangles ABC and DAC:

• ∠ACB = ∠DCA  (common angle)
• ∠BAC = ∠ADC  (both are right angles)

Therefore triangles ABC and DAC are similar.

As they are both similar to ABC, triangles DBA and DAC are similar to each other.  So ratios between corresponding sides are equal, and so:

• BD / AD  =  AD / CD    ⇒    BD × CD = AD²